Question

Mike buys a perpetuity-immediate with varying annual payments. During the first 5 years, the payment is constant and equal to 10. Beginning in year 6, the payments start to increase. For year 6 and all future years, the payment in that year is K% larger than the payment in the year immediately preceding that year, where K < 9.2. 

At an annual effective interest rate of 9.2%, the perpetuity has a present value of 167.50. 

Calculate K

Answer 1

Computation of Discounted cash flows

Year	Dividend	Disc @ 9.2%	DCF
1	10	0.915750916	9.157509
2	10	0.83859974	8.385997
3	10	0.76794848	7.679485
4	10	0.703249523	7.032495
5	10	0.644001395	6.440014
		Total	38.6955

Let the Growth rate in dividend be K%

The PV of future cash flows from year 6 to year infinity = Dividend in 6th year/I%-K%

(10+10*K%)/9.2%-K%

Present value of first 5 years+ Present value of year 6 to infinity cash flows= Present value of the perpetuity =167.5

38.695+(10+k%)/(9.2%-K%)= 167.5

(10+K%)/(9.2%-K%)=167.5-38.695

(10+0.1K)/9.2%-K%=128.805

10+0.1K = (9.2%-K%)*128.805

10+0.1K= 11.85006-1.28805K

1.38805K=1.85006

K% = 1.3328%

Hence K = 1.3328%